2. GRAPES is a software for drawing the
graphs of functions, relations (equation and
inequality), and curves described in
parametric or polar forms; in addition, it
enables you to explore many of their
properties.
3. BASIC STEPS FOR MANIPULATIONS
• Producing graphs or objects
• Operating on graphs
• Appearance of graph
• Tools for analyzing function
• Other useful tools
• Adding caption to a graph or project
• Other supplemental tools and managing
projects
4. Producing graphs or objects
Input and edit function
Input and edit relation
(equation/inequality)
Input and edit inequalities defining regions
Input and edit the expression for curves or
elementary objects (elementary objects :
circle, point, horizontal and vertical lines)
Configuration of line and polygon
(segment, line, line with arrow, rectangle,
angle, line graph, polygon)
5. Operating on graphs
Increase or decrease parameters, and
substitute for parameters
Drag a point
Manipulate an after-image
9. Adding caption to a graph
or project
Display or edit Note
Use of label
10. Other supplemental tools and
managing projects
Print
Copy to clipboard
Save or successive save of images
Initialization of project
Default update
File processing
File association
12. Linear Equations
The "General Form" of the equation of a
straight line is: Ax + By + C = 0
A or B can be zero, but not both at the same
time.
The General Form is not always the most
useful form, and you may prefer to use:
The Slope-Intercept Form of the equation of a
straight line:
y = mx + b
13. Example: Convert 4x - 2y - 5 = 0 to Slope-
Intercept Form
We are heading for
y = mx + b
Start with
4x - 2y - 5 = 0
Move all except y to the left:
-2y = -4x + 5
Divide all by (-2): y = 2x - 5/2
And we are done! (Note: m=2 and b=-5/2)
16. Using the Quadratic Formula
Just put the values of a, b and c into the
Quadratic Formula, and do the calculations.
Example: Solve 5x² + 6x + 1 = 0
Coefficients are:
a = 5, b = 6, c = 1
Quadratic Formula:
x = [ -b ± √(b2-4ac) ] / 2a
Put in a, b and c:
x = [ -6 ± √(62-4×5×1) ] / (2×5)
17. Solve:
x = [ -6 ± √(36-20) ]/10
x = [ -6 ± √(16) ]/10
x = ( -6 ± 4 )/10
x = -0.2 or -1
And we see them on this graph.
18. Example: Solve 5x² + 2x + 1 = 0
Coefficients are:
a = 5, b = 2, c = 1
Note that The Discriminant is negative:
b2 - 4ac = 22 - 4×5×1 = -16
Use the Quadratic Formula:
x = [ -2 ± √(-16) ] / 10
The square root of -16 is 4i
(i is √-1, read Imaginary Numbers to find
out more)
So:x = ( -2 ± 4i )/10
19. Answer: x = -0.2 ± 0.4i
The graph does not cross the x-axis. That is
why we ended up with complex numbers.
In some ways it is easier: we don't need more
calculation, just leave it as -0.2 ± 0.4i.
